The CUSUM Chart, short for Cumulative Sum Control Chart, represents an advanced tool within the Six Sigma methodology, particularly under the broad category of Control Charts. Its primary function is to detect small shifts from the process mean over time, making it an invaluable asset in the realm of quality control and improvement. This article delves into the theory behind CUSUM charts, outlines their construction, and illustrates their practical application within Six Sigma Control Plans.

Theory Behind CUSUM Charts

CUSUM Charts are grounded in the principle of cumulative sums, which involves the aggregation of differences between individual data points and a target value or reference. This approach is particularly sensitive to minor shifts in the process mean, enabling early detection of trends that traditional control charts might overlook. The theory posits that by closely monitoring these cumulative sums, quality control managers can identify deviations from the standard process behavior, signaling the need for corrective actions before the process outputs fall outside acceptable limits.

Construction of CUSUM Charts

Constructing a CUSUM Chart involves several key steps, each critical for its accurate interpretation and effectiveness in detecting process shifts:

1. Determine the Target Value and Control Limits: The target value is typically the process mean or an established standard. Control limits are calculated based on the acceptable level of variation and the desired sensitivity of the chart.

2. Calculate the Cumulative Sum: For each data point, subtract the target value and add this difference to the cumulative sum of the previous data points. This results in two series of cumulative sums, one for positive deviations and another for negative deviations.

3. Plot the CUSUM Chart: On the chart, plot the cumulative sums against the sequence of data points. A separate line for positive and negative deviations may be used to enhance clarity.

4. Interpret the Chart: Key indicators of a process shift include a sustained drift in one direction, crossing of control limits, or patterns that deviate from the expected random fluctuation around the target value.
   

Application in Six Sigma Control Plans

The CUSUM Chart's sensitivity to small process shifts makes it a powerful tool within Six Sigma Control Plans. Its application is particularly suited to processes where the cost of deviations is high, and early detection is critical. Some of the practical applications include:

• Monitoring Manufacturing Processes: For high-precision manufacturing, small shifts can lead to significant quality issues. CUSUM charts help in early detection, reducing scrap rates and improving yield.

• Pharmaceutical Quality Control: Ensuring consistent drug potency and purity can benefit from CUSUM charts by identifying trends that could indicate a drift in the production process.

• Service Industry Performance Tracking: In services, slight variations in process times or customer satisfaction scores can be early indicators of larger issues. CUSUM charts can highlight these trends for early intervention.
  

Conclusion

The CUSUM Chart is a sophisticated analytical tool in the Six Sigma toolkit, designed to detect and monitor small, yet significant, shifts in process performance. Its construction and interpretation require a deep understanding of the process and a careful analysis of the data. When applied correctly within Six Sigma Control Plans, CUSUM Charts can lead to early detection of potential problems, enabling proactive management and continuous process improvement. The ability to identify trends before they result in non-conformities not only enhances process control but also contributes to the overall strategic goal of achieving operational excellence.

Example of CUSUM Chart:

The Cumulative Sum (CU SUM) chart is undeniably intricate, posing challenges in both maintenance and comprehension. Given its complexity, I do not advocate for its routine use in projects. Instead, simpler charting alternatives should be prioritized unless absolutely necessary. Should you find yourself in a situation where a CU SUM chart is indispensable, I strongly recommend leveraging mathematical and statistical software for its creation.

Nevertheless, for the sake of completeness, I will provide an overview. While familiarity with CU SUM charts is essential for the black belt exam, a deep dive into the intricacies of chart construction is not imperative. Understanding the workings and utility of a CU SUM chart suffices.

Due to its complexity, I find it more effective to demonstrate CU SUM charts through an Excel presentation. Hence, I urge you to download the accompanying Excel file and review the example provided therein rather than relying solely on the text below.

By exploring the example in Excel, you'll gain a more practical understanding of CU SUM charts, ensuring a smoother grasp of their functionality and application.

Step 1: Collect the data

Please refer to the Excel provided for a more comprehensive understanding.

Data points represent daily average temperatures (in degrees Celsius) over a 10-day period in a region where the climate is typically stable.

Step2:

Estimate the standard deviation of the data from the moving range control chart σ= Average Moving Range/d2

Average Moving Range : 2.943333333 n=2

d2 =1.128 (from the reference document below)

Let's calculate the standard deviation.

σ= Average Moving Range / D2

σ=2.94333333333333/1.128

σ = 2.609338061

Step 3:

Find value K,

To find the value of K for a CUSUM chart:

Identify Desired Sensitivity: Determine the smallest shift in the process mean you want to detect, typically between 0.5σ and 2σ.

Use Formula: Set K to about half the magnitude of the shift you want to detect, so K=1/2​d, where d is the size of the shift in terms of the process standard deviation (σ).

Adjust Based on Feedback: If the chart is too sensitive or not sensitive enough, adjust K accordingly.

Choose K based on the specific shifts you're interested in detecting and refine it through testing and monitoring the chart's performance.

Calculate the reference value or allowable slack

K= 0.5σ

K =0.5*2.60933806146572

K = 1.304669031

Step 4:

Choose H,

The decision interval H in a CUSUM chart, also known as the control limit, is the threshold value that determines when an out-of-control signal is triggered, indicating a potential shift in the process mean. Here are the key points about H:

Purpose: H sets the sensitivity of the CUSUM chart. A lower H makes the chart more sensitive to small shifts, while a higher H reduces false alarms but may delay detection of actual shifts.

Selection: Choosing H involves balancing the risk of false alarms (Type I error) against the risk of missing a real shift (Type II error). It's often set based on the desired Average Run Length (ARL), which is the expected number of samples before an out-of-control signal is given.

Common Practice: While there's no one-size-fits-all value for H, a common approach is to set H so that the chart has an ARL of 370 to 500 when the process is under control (no shift). For detecting a specific shift size, H might be set to achieve an ARL of around 50 to 100.

Adjustment: The optimal value for H can be determined through simulation studies or using statistical software, based on historical data and the specific context of the process being monitored.

Empirical Rule: Although specific H values depend on the process and the chosen K, an empirical starting point for H is around 4 to 5 times K.

The choice of H is crucial for effective monitoring and should be tailored to the specific detection needs and risk tolerance of the process.

Here we'll choose H = 4

Step 5:

Calculate UCL & LCL

UCL= (H*σ)=

UCL= (4*2.60933806146572)

UCL= 10.43735225

LCL= (H*σ)*(-1)=

UCL= (4*2.60933806146572)*-1

LCL= -10.43735225

Step 6:

Calculate the upper and lower CUSUM values for each individual value.

Target value = 25.202

For the set of temperature data provided, we've set the target value to 25.202°C, reflecting an optimal or expected average daily temperature for a specific environmental or agricultural context. This target could represent the ideal conditions for crop growth, environmental stability, or other temperature-sensitive operations.

Upper CUSUM (UCi)= Max[0, UCi-1+xi – Target value-k)

Lower CUSUM (LCi)= Min[0, LCi-1+xi – Target value+k)

For the first sample the value mesured is : 23.75

Upper CUSUM (UC1)= Max[0, 0+23.75-25.202-1.30466903073286) = 0

Lower CUSUM (UC1)= Min[0, 0+23.75-25.202+1.30466903073286) = -0.147330969 For the second sample the value mesured is : 29.51

Upper CUSUM (UC1)= Max[0,0+29.51-25.202-1.30466903073286) = 3.003330969

Lower CUSUM (UC1)= Min[0,-0.147330969267141+29.51-25.202+1.30466903073286) = 0

Step 7:

Now that I've explained how to calculate the upper and lower CUSUM for the first two values, let's proceed to do the same for all the values in the table below.

It is time to do it for all the value in the table below.

Step 8:

Let's generate the graph from the table.

Here the Excel version:

Here a version from a profesinal software

*as you can see the first point of Lower CUSUM is bugged. It should be at -0.1473

Step 9:

Analyze the CUSUM chart:

The CUMSUM chart analysis indicates a process experiencing initial variability, followed by stabilization, a phase of negative deviations, and a subsequent recovery towards the target. Despite fluctuations, the process remains within control limits, suggesting it is under control without significant issues.